The distance, c = AB, across a lake can be determined by knowing mangle ∠ C, a, and b. Determine the distance AB given mangle ∠ C = 85degrees°, a = 88, and b = 62. Round to two decimal places.

To determine the distance \( AB = c \) across the lake using the given values \( \angle C = 85^\circ \), \( a = 88 \), and \( b = 62 \), you can use the **Law of Cosines**.

The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

### Step-by-Step Calculation:

1. **Substitute the given values into the formula:**

\[
c^2 = 88^2 + 62^2 - 2 \times 88 \times 62 \times \cos(85^\circ)
\]

2. **Calculate each term:**

\[
88^2 = 7744
\]
\[
62^2 = 3844
\]
\[
2 \times 88 \times 62 = 10912
\]
\[
\cos(85^\circ) \approx 0.0872
\]

3. **Substitute these into the equation:**

\[
c^2 = 7744 + 3844 - 10912 \times 0.0872
\]
\[
c^2 = 11588 - 951.96
\]
\[
c^2 = 10636.04
\]

4. **Take the square root of both sides to find \( c \):**

\[
c = \sqrt{10636.04} \approx 103.13
\]

### Final Answer:
The distance \( AB = c \) across the lake is approximately **103.13 units**.

Would you like more details or have any questions?

---

Here are 5 related questions that might interest you:

1. How do you apply the Law of Cosines for an obtuse angle?
2. What is the difference between the Law of Cosines and the Law of Sines?
3. Can the Law of Cosines be used to determine angles?
4. How do changes in angles affect the distance calculation using the Law of Cosines?
5. What are some real-world applications of the Law of Cosines?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when dealing with trigonometric functions to avoid errors.

The distance, c = AB, across a lake can be determined by knowing ∠C, a, and b. Determine the distance AB given ∠C = 85°, a = 88, and b = 62. Round to two decimal places.

Learn how to determine the distance across a lake using the Law of Cosines, given angle ∠C = 85°, a = 88, and b = 62. This step-by-step solution demonstrates key trigonometric concepts, perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation